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Method for finding mechanism and activation energy of magnetic transitions, applied to skyrmion and antivortex annihilation

Pavel F. Bessarab, Valery M. Uzdin, Hannes Jónsson

arXiv:1502.05065v1cond-mat.mes-hall

TL;DR

The paper addresses how to identify mechanisms and activation energies for complex magnetic transitions, where locating relevant saddle points is difficult. It presents geodesic nudged elastic band (GNEB), which accounts for the curvature of magnetic configuration space, and applies it to skyrmion and antivortex transitions. The calculations illustrate GNEB’s applicability to complex magnetic systems and support lifetime estimation for an antivortex state, subject to assumptions about magnetic-moment magnitudes.

  • Problem

    A methodology is needed to accurately determine the mechanism and rate of annihilation of noncollinear magnetic states, while relevant saddle points are difficult to locate in systems with many degrees of freedom.

  • Method

    GNEB finds minimum energy paths by using geodesics for distances and displacements and projecting path tangents and magnetic forces onto the orientation manifold.

  • Results

    GNEB calculations of skyrmion and antivortex annihilation illustrate its applicability to complex magnetic transitions and provide a basis for determining the antivortex lifetime within harmonic transition state theory.

  • Takeaways & Limitations

    Accounting for configuration-space curvature provides a tool for studying more complex magnetic systems than could be studied with the NEB method.

  • Takeaways & Limitations

    The magnitude of the magnetic moments is assumed to adjust with orientation, either through a fixed-magnitude treatment or separate self-consistency calculations.

Abstract

from arXiv · show

A method for finding minimum energy paths of transitions in magnetic systems is presented and used to determine the mechanism and estimate the activation energy of skyrmion and antivortex annihiliation in nano-systems. The path is optimized with respect to orientation of the magnetic vectors while their magnitudes are fixed or obtained from separate calculations. The curvature of the configuration space is taken into account by: (1) using geodesics to evaluate distances and displacements of the system during the optimization, and (2) projecting the path tangent and the magnetic force on the tangent space of the manifold defined by all possible orientations of the magnetic vectors. The method, named geodesic nudged elastic band (GNEB), and its implementation are illustrated with calculations of complex transitions involving annihilation and creation of skyrmion and antivortex states. The lifetime of the latter was determined within harmonic transition state theory using a noncollinear extension of the Alexander-Anderson model.

I. INTRODUCTION

Thermal stability of noncollinear magnetic states matters for information technologies, but identifying their transition mechanisms and activation barriers is difficult. The paper introduces GNEB to find minimum energy paths for complex magnetic transitions.

  • Motivation: Thermally activated transitions can destroy magnetic states, making activation energy the primary quantity determining thermal stability.The activation energy is the energy difference between the initial-state minimum and the highest first-order saddle point on a connecting path.
  • Limitations of existing approaches: Finding relevant saddle points is challenging because the calculation must identify which degree of freedom acts as the reaction coordinate.Preselected reaction coordinates can be inaccurate, produce discontinuous paths, and yield unreliable saddle-point energies.
  • Minimum energy paths: A minimum energy path connects initial and final minima while minimizing energy in directions perpendicular to the path, thereby representing a transition mechanism.Its highest maximum estimates the activation energy, and multiple paths can correspond to different mechanisms and rates.
  • Limitations of existing approaches: Existing approaches can converge to higher-order saddle points or local maxima instead of a minimum energy path, especially as system dimensionality grows.Such false paths do not provide a thermal activation-energy estimate.
  • Contribution: GNEB extends NEB to complex magnetic transitions involving three-dimensional rotations by accounting for configuration-space curvature and projecting the path tangent and forces.The method is applied to skyrmion and antivortex annihilation and to complex magnetic transitions.

II. NEB METHOD

NEB represents a transition as a chain of images and iteratively moves intermediate images toward a minimum energy path. Force projection preserves path resolution near saddle points while springs control image spacing.

  • Path representation: NEB uses a chain of Q images with fixed endpoint configurations and Q −2 intermediate images representing a discrete transition path.The intermediate images are adjusted iteratively to converge on the minimum energy path.
  • Force projection: Only transverse components of the true force move images, while the spring force is projected along the path to control image distribution.This prevents images from sliding toward low-energy regions and losing information near saddle points.
  • Image distribution: Image density should be increased where energy is high or path curvature is large, because some path regions require higher resolution.Insufficient control over image distribution can reduce density near saddle points, where resolution is especially important.
  • Tangent estimation: A tangent estimated from the higher-energy neighboring image improves stability relative to a simple segment connecting the previous and next images.The simple estimate can cause instabilities that slow or prevent convergence.
  • Climbing image: The climbing image removes the spring force and reverses the parallel true-force component, moving uphill along the path and downhill perpendicular to it.After convergence, it coincides with the highest saddle point and gives an accurate saddle-point energy.

III. GEODESIC NEB METHOD

GNEB reformulates nudged elastic band optimization on the curved manifold of magnetic-moment orientations. It uses geodesic geometry and tangent-space projections to maintain valid constraints and stable image distributions.

  • Physical assumptions: The method assumes magnetic-moment magnitudes are fixed or determined separately, while optimizing their orientations.Changes in magnitude are treated as faster than changes in orientation.
  • Configuration space: GNEB performs unconstrained optimization on a 2N-dimensional Riemannian manifold formed by the direct product of N unit spheres.This replaces 3N-dimensional Euclidean coordinates with N magnitude constraints.
  • Path optimization: The method represents transitions with a chain of images connected by springs between fixed initial and final minima.Intermediate images are adjusted until the GNEB force is zero.
  • Force projection: Projecting both the path tangent and force onto the manifold tangent space decouples true and spring forces while satisfying magnetic constraints.Without tangent projection, force interference can produce uncontrolled image behavior.
  • Geodesic geometry: Geodesic distances evaluate image separations and spring forces on the curved configuration manifold.Each manifold distance combines great-circle distances on the individual unit spheres.

A. Test problems

Single-spin tests show that tangent-space projection and springs are essential for reliable GNEB convergence and path resolution. Correct projection makes results stable across a wide range of spring constants.

  • Numerical robustness: The minimization converged without problems even when a minimum coincided with a pole.The test specifically assessed robustness near the coordinate singularities of the spherical representation.
  • Single-spin MEP: The initial geodesic path between minima is not generally optimal because it can pass close to an energy maximum.GNEB iteratively brings the images from this initialization toward the MEP.
  • Effect of tangent projection: Without tangent projection, images slide from the barrier toward minima, losing resolution near the saddle point.After 400 steps, most images had entered the potential wells and saddle-region information was lost.
  • Effect of tangent projection: With the projected tangent, GNEB is stable and produces nearly identical results across five orders of magnitude in spring constant.The projected formulation is therefore insensitive to the precise spring-constant value in this test.
  • Effect of springs: Springs preserve an approximately equal image distribution and convergence to the MEP, whereas omitting them produces uneven spacing and instability.Without springs, images slide toward the energy minima and become sensitive to small perturbations.

B. Annihilation of a skyrmion

For a lattice skyrmion, CI-GNEB resolves an annihilation pathway that combines nearly barrier-free translation with symmetric spin rotation and shrinkage. The calculated barriers are asymmetric between annihilation and formation.

  • Stable states: Random-spin minimization found a collinear global minimum and an isolated noncollinear metastable skyrmion state.The skyrmion center coincides with a lattice site.
  • Mechanism: The MEP shows the skyrmion first translating between atomic sites with almost no energy change, then shrinking through symmetric spin rotations.The final stage involves out-of-plane rotation before the skyrmion disappears.
  • Activation energy: 40 meV is the activation energy for skyrmion annihilation, versus 50 meV for formation from the collinear state.The saddle configuration has four central spins in the XY-plane pointing toward one another.

C. Annihilation of an antivortex

CI-GNEB identifies antivortex annihilation as motion of a saddle-like excitation toward an island corner and calculates both barriers and harmonic-transition-state prefactors. The reverse transition has a larger prefactor.

  • Antivortex state: The metastable antivortex has a symmetric saddle-like arrangement with zero total in-plane magnetization and nonzero out-of-plane magnetization.Four central moments primarily produce the out-of-plane component.
  • Model context: A 7×7 Fe island can support an antivortex state when model parameters differ slightly from optimal ambient-condition Fe-crystal values.Such differences may arise from external perturbations, impurities, or defects.
  • Mechanism: Along the MEP, the saddle-like excitation travels diagonally toward a corner and exits the island.The transition involves out-of-plane rotation of magnetic-moment vectors.
  • Activation energy: The antivortex activation energy is 10 meV for annihilation and 55 meV for formation from the collinear state.The maximum-energy configuration was verified as a saddle point.
  • Transition-state theory: Harmonic transition state theory gives a 1.4×10^13 s^-1 prefactor for annihilation and 2.6×10^13 s^-1 for the reverse transition.The larger reverse prefactor is attributed to greater vibrational entropy of the antivortex state.

V. DISCUSSION

The GNEB method accounts for the curvature of magnetic configuration space and resolves convergence issues encountered with NEB, enabling complex magnetic-transition studies. Its applications illustrate annihilation mechanisms and support conclusions about model-dependent magnetic states and transition pathways.

  • Scope and limitations: The reported annihilation results remain preliminary because they were obtained for only particular systems and Hamiltonians.The authors question whether similar transition mechanisms will occur in other models supporting these states.
  • Model-dependent states: The NCAA model supports a local minimum corresponding to an antivortex state, while complex noncollinear states arise naturally in the model.Whether the model can support skyrmion states remains unresolved.
  • Applications: The calculations of skyrmion and antivortex annihilation illustrate GNEB’s applicability to complex magnetic transitions.The results are presented mainly to illustrate possible transition mechanisms and provide an indication of thermal stability in simple systems.
  • Method and convergence: The extension resolves convergence issues and provides a tool for studying more complex magnetic systems than NEB could address.Simpler NEB implementations without tangent projection failed to reach convergence for the skyrmion and antivortex calculations.
  • Method and convergence: GNEB accounts for configuration-space curvature through tangent-space projection, a distinction from NEB that is important for rigorous convergence.The method also uses geodesics to represent distances and displacements along the path.

Appendix A: Evaluation of path tangent

The path tangent is estimated from neighboring images, with an energy-dependent weighted scheme used to reduce kinks and convergence instabilities.

  • The tangent at an image is estimated from the coordinates of adjacent images.
  • Naive tangent choices can create kinks that slow or prevent convergence to the minimum energy path.
  • Forward or backward differences are selected according to the image energy.
  • Near energy minima or maxima, the tangent is formed as a weighted average of neighboring tangent estimates.

Appendix B: Evaluation of the geodesic distance

The appendix develops geodesic distances and path interpolation for magnetic images, then uses them to construct, relax, and analyze a minimum energy path.

  • GNEB evaluates distances between magnetic images using geodesic distances on the orientation manifold.
  • The spherical law of cosines is simple but can suffer significant round-off errors for small distances.
  • The haversine formula is effective for small and large distances, while Vincenty’s formula was used in the presented calculations.
  • The iterative path calculation starts from two minima, creates intermediate images, computes energies, projects forces, and updates images until convergence.
  • The total force combines the transverse projected true force with the parallel spring force.
  • A cubic interpolation using image energies and path-gradient components locates saddle points and intermediate minima between images.

Appendix F: Iterative optimization algorithm

The VPO algorithm minimizes energy on the curved magnetic manifold by retaining force-aligned motion and advancing along geodesics rather than straight Euclidean displacements.

  • VPO advances the system along geodesics, automatically satisfying magnetic constraints during optimization.
  • Standard force-directed iterative algorithms are inefficient on curved manifolds because straight-line displacements leave the constrained manifold.
  • The method retains only the velocity component parallel to the force and zeros the velocity when motion overshoots.
  • For N magnetic moments, the optimization uses 2N coupled equations in spherical coordinates for the orientations and their velocities.
  • When the force keeps pointing in a similar direction, the system accelerates, analogous to increasing the timestep in steepest descent.
  • VPO typically converges to the nearest local energy minimum; sampling multiple initial guesses can find the global minimum in principle.

2. Treatment of the poles

Near the poles, spherical coordinates become singular because the azimuthal angle is irrelevant, so the optimization switches to Cartesian coordinates.

  • Near a pole, the azimuthal angle becomes irrelevant and spherical-coordinate VPO breaks down.
  • When θ_i < ϵ or π − θ_i < ϵ, the method uses x and y components of the magnetic moments instead.
  • Force and velocity vectors are transformed into Cartesian coordinates while the system remains within the pole neighborhood.
  • The representative trajectory develops a cusp when the velocity–force inner product becomes negative and the velocity is zeroed.
  • The trajectory initially advances slowly, then accelerates, and converges to a minimum located at the pole.

Appendix G: Verification of a saddle point

The appendix describes how to verify that the maximum-energy point on a converged MEP is a first-order saddle point and how to address higher-order saddle points. It recommends Hessian eigenvalue analysis and perturbations when symmetry or path geometry causes incorrect convergence.

  • Saddle-point verification: This saddle-point analysis is required before evaluating transition rates with harmonic transition-state theory or Kramers methods.The maximum-energy image may be obtained by path interpolation or by the CI-GNEB procedure.
  • Higher-order saddle points: Highly symmetric initial paths can converge to higher-order saddle points, so adding small random noise to the initial path is generally recommended.For simple systems the energy surface and MEP can be visualized directly, while complex magnetic systems require more systematic verification.
  • Path verification: When a ridge continues directly from an energy valley, NEB may converge partly along the ridge rather than along the intended minimum-energy path.This path-geometry issue is another reason to inspect the converged result rather than relying only on convergence.
  • Saddle-point verification: The verification procedure computes the 2N×2N Hessian matrix of energy second derivatives and examines its eigenvalues.Any convenient coordinate system, including spherical or stereographic coordinates, can be used for the second-derivative calculation.
  • Saddle-point verification: A first-order saddle point has exactly one negative Hessian eigenvalue, whereas two negative eigenvalues indicate a second-order saddle point.The remaining modes are minima for a first-order saddle point; higher-order saddle points are maxima along two or more modes.
  • Higher-order saddle points: If NEB converges through a second-order saddle point, the highest-energy image can be displaced along the orthogonal negative mode or refined with minimum-mode following.These procedures seek a path that avoids the additional unstable direction.
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